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Math Expert V
Joined: 02 Sep 2009
Posts: 58427
The standard equation of a circle is (x−h)^2+(y−k)^2=r^2, where the  [#permalink]

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Question Stats: 71% (02:01) correct 29% (02:12) wrong based on 203 sessions

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The standard equation of a circle is $$(x−h)^2+(y−k)^2=r^2$$, where the point (h,k) is the center of the circle and r represents the length of the radius. If a particular circle can be represented by the equation $$x^2+y^2−8x+2y=−1$$, what is the radius of that circle?

A. -1
B. 1
C. 2
D. 4
E. 8

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Marshall & McDonough Moderator D
Joined: 13 Apr 2015
Posts: 1683
Location: India
Re: The standard equation of a circle is (x−h)^2+(y−k)^2=r^2, where the  [#permalink]

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5
1
x^2 + y^2 - 8x + 2y = -1
To bring the above equation into its standard form, complete the squares by adding 16 and 1 on both the sides
x^2 - 8x + 16 + y^2 + 2y + 1 = -1 + 16 + 1
(x - 4)^2 + (y + 1)^2 = 16 --> This is in the form of (x-h)^2 + (y-k)^2 = r^2
r = 4

General Discussion
Intern  Joined: 14 Mar 2014
Posts: 2
Re: The standard equation of a circle is (x−h)^2+(y−k)^2=r^2, where the  [#permalink]

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Hi Bunuel,

This is my first time participating in gmatclub. Firs of all thank you for always posting stuff that really helps and for always giving a good solution to math problems.

I've been struggling trying to find out an easy procedure for this question. Can you help?

Thanks again

Juan
CEO  S
Joined: 20 Mar 2014
Posts: 2595
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44 GPA: 3.7
WE: Engineering (Aerospace and Defense)
Re: The standard equation of a circle is (x−h)^2+(y−k)^2=r^2, where the  [#permalink]

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2
betancurj wrote:
Hi Bunuel,

This is my first time participating in gmatclub. Firs of all thank you for always posting stuff that really helps and for always giving a good solution to math problems.

I've been struggling trying to find out an easy procedure for this question. Can you help?

Thanks again

Juan

Let me try to answer.

You are given that the standard equation of a circle ---> (x−h)^2+(y−k)^2=r^2 where, r=radius of the circle.

Now, you are given another circle with the equation ---> x^2 + y^2 - 8x + 2y = -1 , when you compare this to the standard equation of the circle, you should realize that it would be straightforward to calculate the radius of the circle if you can express x^2 + y^2 - 8x + 2y = -1 in the form of (x−h)^2+(y−k)^2=r^2

In order to make complete squares similar to (x-h)^2 or (y-k)^2, you need to remember that (x+a)^2=x^2+a^2+2ax and (x-a)^2=x^2+a^2-2ax

With these relations in mind, lets go back to x^2 + y^2 - 8x + 2y = -1 with a focus on converting x^2 + y^2 - 8x + 2y to 2 complete squares (1 with x and 1 with y).

x^2 + y^2 - 8x + 2y ---> (x^2-8x+16-16)+(y^2+2y+1-1) = (x-4)^2 -16 + (y+1)^2-1 ---> x^2 + y^2 - 8x + 2y = -1 can thus be written as :

(x-4)^2 -16 + (y+1)^2-1 = -1 --->(x-4)^2 + (y+1)^2 = 16 = 4^2.

NOw you can compare this form to (x−h)^2+(y−k)^2=r^2 to get h = 4, k=-1 and r=radius = 4.

Thus, D is the correct answer.

For questions such as these you need to move backwards from what has been given to you and to modify the given values into a more usable form.

Hope this helps.
SVP  B
Joined: 06 Nov 2014
Posts: 1873
The standard equation of a circle is (x−h)^2+(y−k)^2=r^2, where the  [#permalink]

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1
2
betancurj wrote:
Hi Bunuel,

This is my first time participating in gmatclub. Firs of all thank you for always posting stuff that really helps and for always giving a good solution to math problems.

I've been struggling trying to find out an easy procedure for this question. Can you help?

Thanks again

Juan

Hi Juan,

The trick here is to convert everything in the standard form of equation of a circle, which is:
$$(x−h)^2+(y−k)^2=r^2$$

We are given the equation of a circle in some other form (expanded form of the standard equation)

$$x^2+y^2−8x+2y=−1$$
On grouping terms,

$$x^2−8x+y^2+2y=−1$$
From here on, we need to make two perfect squares in x and y

The best way to understand which perfect square to make is by dissecting the coefficient of x and y
8 = 2*1*4, 2 = 2*1*1

We are writing the coefficients in this form, because the coefficient of x in $${(x+a)}^2 = x^2 + 2ax + a^2$$ (Eq i) is 2*1*a
$$x^2−2*x*4 + y^2+2*1*y = -1$$

Now we need the constant terms to make these perfect squares.
The constant term in Eq (i) is a

Hence we can write $$x^2−2*1*4x + y^2+2*1*y = -1$$ as
$$x^2−2*x*4 + 16 + y^2+2*1*y + 1 = -1 + 16 + 1$$ (adding constants on both sides of the equation)
$${(x-4)}^2 + {(y+1)}^2 = 16$$
$${(x-4)}^2 + {(y+1)}^2 = 4^2$$

Hence the radius = 4. Option D
Intern  Joined: 14 Mar 2014
Posts: 2
Re: The standard equation of a circle is (x−h)^2+(y−k)^2=r^2, where the  [#permalink]

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Thank you both, great help.
Director  S
Joined: 12 Nov 2016
Posts: 699
Location: United States
Schools: Yale '18
GMAT 1: 650 Q43 V37 GRE 1: Q157 V158 GPA: 2.66
Re: The standard equation of a circle is (x−h)^2+(y−k)^2=r^2, where the  [#permalink]

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Bunuel wrote:
The standard equation of a circle is $$(x−h)^2+(y−k)^2=r^2$$, where the point (h,k) is the center of the circle and r represents the length of the radius. If a particular circle can be represented by the equation $$x^2+y^2−8x+2y=−1$$, what is the radius of that circle?

A. -1
B. 1
C. 2
D. 4
E. 8

Bunuel -

I simply factored (x-h)^2 + (y-k)^2 = r^2

[x^2-2xh + h] + [y^2-2yk + y^2] = r^2

If it can be reduced to

x^2 +y^2 -8x +2y -1

Then don't H and K have to be 4 and -1? Unless I'm making some assumption here that wouldn't necessarily apply to some other example- please let me know

If (x-4)^2 + (y - (-1))^2 = r^2
(4-4)^2 + (1 +1)^2 = r^2
(2)^2= r^2
4= r^2
4 = \sqrt{r}
4= r
Director  S
Joined: 12 Nov 2016
Posts: 699
Location: United States
Schools: Yale '18
GMAT 1: 650 Q43 V37 GRE 1: Q157 V158 GPA: 2.66
Re: The standard equation of a circle is (x−h)^2+(y−k)^2=r^2, where the  [#permalink]

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OptimusPrepJanielle wrote:
betancurj wrote:
Hi Bunuel,

This is my first time participating in gmatclub. Firs of all thank you for always posting stuff that really helps and for always giving a good solution to math problems.

I've been struggling trying to find out an easy procedure for this question. Can you help?

Thanks again

Juan

Hi Juan,

The trick here is to convert everything in the standard form of equation of a circle, which is:
$$(x−h)^2+(y−k)^2=r^2$$

We are given the equation of a circle in some other form (expanded form of the standard equation)

$$x^2+y^2−8x+2y=−1$$
On grouping terms,

$$x^2−8x+y^2+2y=−1$$
From here on, we need to make two perfect squares in x and y

The best way to understand which perfect square to make is by dissecting the coefficient of x and y
8 = 2*1*4, 2 = 2*1*1

We are writing the coefficients in this form, because the coefficient of x in $${(x+a)}^2 = x^2 + 2ax + a^2$$ (Eq i) is 2*1*a
$$x^2−2*x*4 + y^2+2*1*y = -1$$

Now we need the constant terms to make these perfect squares.
The constant term in Eq (i) is a

Hence we can write $$x^2−2*1*4x + y^2+2*1*y = -1$$ as
$$x^2−2*x*4 + 16 + y^2+2*1*y + 1 = -1 + 16 + 1$$ (adding constants on both sides of the equation)
$${(x-4)}^2 + {(y+1)}^2 = 16$$
$${(x-4)}^2 + {(y+1)}^2 = 4^2$$

Hence the radius = 4. Option D

Yes but don't forget to explain you derived 16 and 1 by taking half of the coefficients 8 and 1 and then squaring them. For example, if we have an algebraic expression that cannot be factored such as x-10x-18=0 - then we complete the square. How?

x-10x __ = 18 (add 18 to both sides or really just put 18 on the other side)
x-10x +25 = 18 + 25 (remember how to get the coefficients? In this equation a= 1 b= 10; take (b/2)^2 and add to both sides
(x-5)(x-5) = 43
x-5 = + or minus 43
Math Expert V
Joined: 02 Sep 2009
Posts: 58427
Re: The standard equation of a circle is (x−h)^2+(y−k)^2=r^2, where the  [#permalink]

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Bunuel wrote:
The standard equation of a circle is $$(x−h)^2+(y−k)^2=r^2$$, where the point (h,k) is the center of the circle and r represents the length of the radius. If a particular circle can be represented by the equation $$x^2+y^2−8x+2y=−1$$, what is the radius of that circle?

A. -1
B. 1
C. 2
D. 4
E. 8

VERITAS PREP OFFICIAL SOLUTION:

What should first jump out is that the equation for this particular circle does not match the standard equation of a circle. So you'll need to do some algebra to get to that point.

Your first step should be to factor the x and y terms so that they fit the standard form. Since each term needs to be factored into a squared parenthetical, you should use the common quadratic form $$(a−b)^2=a^2−2ab+b^2$$ as your guide. Look at the coefficients −8x and +2y, recognizing that they need to fill the −2ab role in that common quadratic equation. This means that you'll factor the x and y terms into:

$$(x−4)^2=x^2−8x+16$$
and
$$(y−(−1)^2)=y^2+2y+1$$

So now you know that the left-hand side of the standard equation of a circle produces the combination of $$x^2−8x+16+y^2+2y+1$$, which equals $$x^2−8x+y^2+2y+17$$. So now you know two equations to be true:

$$x^2−8x+y^2+2y+17=r^2$$

and

$$x^2+y^2−8x+2y=−1$$

That second equation can be quickly made to look like the first by adding 1 to both sides:

$$x^2+y^2−8x+2y+1=0$$

Which means that the difference between the two equations is that the left-hand side is 16 apart and the right-hand side is r^2 apart. So to balance, add 16 to the left and r^2 to the right, meaning that r^2=16. This means that the radius is 4, making answer choice D correct.
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Re: The standard equation of a circle is (x−h)^2+(y−k)^2=r^2, where the  [#permalink]

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